Theorem nrexdv 2570

Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Hypothesis

Ref	Expression
nrexdv.1	|- ( ( ph /\ x e. A ) -> -. ps )

Assertion

Ref	Expression
nrexdv	|- ( ph -> -. E. x e. A ps )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem nrexdv

Step	Hyp	Ref	Expression
1		nrexdv.1	. . 3 |- ( ( ph /\ x e. A ) -> -. ps )
2	1	ralrimiva 2550	. 2 |- ( ph -> A. x e. A -. ps )
3		ralnex 2465	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	sylib 122	1 |- ( ph -> -. E. x e. A ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2148   A.wral 2455   E.wrex 2456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-17 1526

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-ral 2460  df-rex 2461

This theorem is referenced by:  ltpopr  7593  cauappcvgprlemladdru  7654  cauappcvgprlemladdrl  7655  caucvgprlemladdrl  7676  caucvgprprlemaddq  7706  dvdsle  11849